Critical Temperature of Periodic Ising Models

TL;DR

This paper precisely characterizes the critical temperature of periodic Ising models using spectral curve analysis of associated dimer models, providing new insights into phase transition behavior and correlation decay.

Contribution

It offers an exact algebraic condition for the critical temperature of periodic Ising models via spectral curve analysis, linking it to dimer model properties.

Findings

Exact algebraic formula for critical temperature.

Exponential decay of correlations above critical temperature.

Connection between Ising model criticality and dimer spectral curve.

Abstract

A periodic Ising model is one endowed with interactions that are invariant under translations of members of a full-rank sublattice $\mathfrak{L}$ of $\mathbb{Z}^2$. We give an exact, quantitative description of the critical temperature, defined by the supreme of the temperatures at which the spontaneous magnetization of a periodic, Ising ferromagnets is nonzero, as the solution of a certain algebraic equation, namely, the condition that the spectral curve of the corresponding dimer model on the Fisher graph has a real node on the unit torus. A simple proof for the exponential decay of spin-spin correlations above the critical temperature for the symmetric, periodic Ising ferromagnet, as well as the exponential decay of the edge-edge correlations for all non-critical edge weights of the dimer model on periodic Fisher graphs, is obtained by our technique.